We recently added an enhancement to the chatroom Rasa widget here and we documented it’s use in the example helpdesk-assistant here.
It is an example of what you show above without the use of a Router or Handoff DB. When the user requests a transfer, the Rasa response can include slots or metadata that is passed to the UI component. In the example implementation, it passes the URL to another Rasa chatbot but you could add your own metadata.